Question: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{4t^2 - 32t + 48}{9t^2 - 135t + 486}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {4(t^2 - 8t + 12)} {9(t^2 - 15t + 54)} $ $ z = \dfrac{4}{9} \cdot \dfrac{t^2 - 8t + 12}{t^2 - 15t + 54} $ Next factor the numerator and denominator. $ z = \dfrac{4}{9} \cdot \dfrac{(t - 6)(t - 2)}{(t - 6)(t - 9)}$ Assuming $t \neq 6$ , we can cancel the $t - 6$ $ z = \dfrac{4}{9} \cdot \dfrac{t - 2}{t - 9}$ Therefore: $ z = \dfrac{ 4(t - 2)}{ 9(t - 9)}$, $t \neq 6$